This article is meant for readers who, like me, have studied Linear Algebra and who, like me, are curious about Quantum Mechanics.
Are the columns of matrices in a given, natural order, as we write them? Well, if we are using the matrix as a rotation matrix in CGI – i.e. its elements are derived from the trig functions of Euler Angles – then the column order depends, on the order in which we have labeled coordinates to be X, Y and Z. We are not free to change this order in the middle of our calculations, but if we decide that X, Y and Z are supposed to form a different set, then we need to use different matrices as well.
(Edited 02/15/2018 :
OTOH, We also know that a matrix can be an expression of a system of simultaneous equations, which can be solved manually through Gauss-Jordan Elimination on the matrix. If we have found that our system has infinitely many solutions, then we are inclined to say that certain variables are the “Leading Variables” while the others are the “Free Variables”.
It is being taught today, that the Free Variables can also be made our parameters, so that the set of values for the Leading Variables follows from those parameters. But wait. Should it not be arbitrary for certain combinations of variables, which follows from which?
The answer is, that if we simply use Gauss-Jordan Elimination, and if two variables are connected as having possibly infinite combinations of values, then
it will always be the variables stated earlier in the equations which become the Leading, and the ones stated later in the equations will become the Free Variables. We could restate the entire equations with the variables in some other order, and then surely enough, the variable that used to be a Free one will have become a new Leading one, and vice-versa. (And if we do so, the parametric equations for the other Leading variables will generally also change.)
As of 02/15/2018:
This is an observation which I once made, based on certain exercises in Linear Algebra, as taught, having been simplified in the way I described. Eventually, systems of solutions will come up in the real world, in which a Free Variable actually precedes a Leading Variable, both in the order they get mentioned in equations, as well as according to the order of matrix-columns. The corresponding row with a single 1, corresponding to those Free Variables, will not occur, so that it also cannot be added to or subtracted from what will be the earlier row, in such a case.
The later observation follows from the fact that such solutions have infinitely many solutions:
- If the Free Variables are just given a set of values that follow from the solution-set, Then the Leading variables would still need to have definite values, as defined by the same solution-matrix,
- If the Free Variables are given a set of values, that no longer follow from the solution-set, it will not follow that a different set of values for the Leading Variables, will make such extraneous solutions viable.
End of edit, 02/15/2018 )
The order of the columns, has become the order of discovery.
This could also have ramifications for Quantum Mechanics, where matrices are sometimes used. QM used matrices at first, in an effort to be empirical, and to acknowledge that we as Humans, can only observe a subset of the properties which particles may have. And then what happens in QM, is that some of the matrices used are computed to have Eigenvalues, and if those turn out to be real numbers, they are also thought to correspond to observable properties of particles, while complex Eigenvalues are stated – modestly enough – not to correspond to observable properties of the particle.
Even though this system seems straightforward, it is not foolproof. A Magnetic North Pole corresponds according to Classical Principles, to an angle from which an assumed current is always flowing arbitrarily, clockwise or counter-clockwise. It should follow then, that from a different perspective, a current which was flowing clockwise before, should always be flowing counter-clockwise. And yet according to QM, monopoles should exist.
The reason this can happen according to QM, is the fact that in QM, nobody is shy to apply imaginary numbers, and imaginary numbers can be applied in such a way, that regardless of which angle we view a monopole from, its electrical current will always be flowing clockwise, say.
But, since the same QM architects must be applying the principle, that complex – i.e. imaginary – Eigenvalues do not form observable properties, it must follow that according to their matrices, this electrical current imagined in Classical E&M, does not correspond to a fundamental property. If it did, the fact that this could become complex, would also lead to a non-observable property…
Well you see, here is where I see a problem. The order in which QM has columns in their matrices today, is actually likely to have a strong chronological ordering. The first columns are likely to correspond, to properties that have been discovered maybe 400 years ago, and that have been thought fundamental ever since. The last columns are likely to correspond to properties that were discovered very recently.
And so it would seem that there is some property which our ancestors considered fundamental 400 years ago, which still form part of the matrices which get used in QM today. Just because these are fundamental in a macroscopic examination of our Universe, in my opinion, does not prove, that they are also intrinsic to fundamental particles.
What happens with matrices however, is that their base-vectors form a type of coordinate system, which can be altered, so that a different coordinate system will require a different matrix, to state the same thing, effectively, that the earlier matrix stated. This tends to be correct as long as the number of coordinates is correct, and as long as the matrix which transforms from one system to the next does not have a determinant of zero. As long as the determinant is not zero, we can translate from one system of variables to the other, and by definition back again, because such matrices also have definable inverses.
This observation is what led me to offer This Earlier Posting, in which I stated that a full diagonalization was called for. Granted, we have accepted complex numbers into the realm of Physics already. Would it not make sense then, to try to compute a Perpendicular Matrix – i.e. an Orthogonal Basis – such that all the complex numbers are in this Perpendicular Matrix, but such that the Diagonal Matrix which results, will only have real numbers?
I suppose that an important question to ask would be, ‘Then what? Are we supposed to inspect these Perpendicular Matrices, to find some obvious symmetry which we have missed before? That flies into our human eyes, since Humans understand Complex Algebra so well?’ I suppose the answer is No. But then at least, the Perpendicular Matrix will translate Classical Properties to and from this Diagonal Matrix, and we could then know that a set of properties exists, which should in principle be observable. But which we have yet to observe.
(Edit : ) I should give two hypothetical examples:
- Electrical Current can exist in a secondary way, when a swarm of particles that do have Charge, move across a boundary. This does not prove, that Current is a fundamental property of particles however.
- Certain Linear properties are commonly thought just to have an Angular homologue. For example, Angular momentum and force could exist on the macroscopic level as a kind of Integral, because individual particles have the Linear property, but in opposing directions, on opposite sides of a center of mass. This does not prove, that torque is truly a fundamental property for particles, nor spin. Yet, I am aware that a particle is stated to have a Spin Quantum Number. Which is Not a Vector. I can doubt that this number, truly has anything to do with the type of angular momentum which was thought to be basic, 400 years ago.
400 Years ago, macroscopic objects were known to be able to spin along an axis, that was not their axis of linear travel. According to QM however, the spin vector of a particle is assumed to be parallel to the momentum vector. Why then, If Angular Momentum is just the Quantum-Mechanical homologue, as macroscopic Angular Momentum was?
One possible answer could be, that it is just so. Another possible answer could be, that integers are a subset of real numbers, that spin should be an observable property, but that inserting this quantum number completes a matrix, such that a complex Eigenvalue results someplace else.
(Edit 09/13/2016 : ) I have 2 more observations about this sort of diagonal matrix:
- It would not be a magic bullet, to find new properties of particles. This is because some if its elements will correspond directly to Eigenvalues already known to be real, as its others will, to Eigenvalues already known to be complex. This approach will just insist on solving for both varieties. And none of the diagonal elements can be equal to zero, as this would also imply a determinant of zero, as well as zero being one of the Eigenvalues, with its own Eigenspace.
- If spin was really just a secondary phenomenon as suggested above, then it would be difficult to explain why it is also Universally conserved.